1.

For a positive integer n show that (1+i)^n+(1-i)^n=2^((n+2)/2) "cos((npi)/4)

Answer»

Solution :`L.H.S.=(1+i)^n+(1-i)^n`
`={sqrt2(1/sqrt2^1+i1/sqrt2)}^n+{sqrt2(1/sqrt2-i1/sqrt2)}^n`
`=2^(n//2)[("cos"pi/4="isin"pi/4)^n="cos"pi/4 - "isin"pi/4)^n]`
`=2^(n//2)xx2"cos"(NPI)/4=2^(n//2+1)"cos"(npi)/4`
`=2^((n+2)/2) "cos"(npi)/4=R.H.S`


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